Grand Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of Bose–Einstein Condensation of an Ideal Gas in Arbitrary Trap

Tarasov, S. V.; Kocharovsky, Vl. V.; Kocharovsky, V. V.

doi:10.1007/s10955-015-1361-3

Cited by 14 publications

(20 citation statements)

References 81 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where x ∈ R + , x 1 and ∼ refers to asymptotically in m. When x is any positive real, not necessarily large m-independent number, Erdös and Lehner [39] report a more precise result as a function of every x, which has been exploited by others [4][5][6]15,24] to predict the behavior of some Bosonic systems at low temperature. The above result (7) and (8) predates that, in Equation (6); however, it can be interpreted by comparison with Equation (6), if we suppose that, asymptotically in m, the distribution of r values in the partitions counted by p(m) is sharply peaked around its maximum. Therefore, r(m) defined in Equation (6) is proximate to the optimal number of summands.…”

Section: Partitions and Compositions: The Case Of Integerssupporting

confidence: 63%

“…This is already widely covered even in textbooks on the subject, thus we only touch upon it. There are, however, plenty of results in the mathematical community and in number theory dealing with asymptotic (mainly large m) estimations of the number of partitions or compositions of an integer into a sequence of other assigned integers [25][26][27][28][29][30][31][32][33], many of which have direct physical applications [4][5][6][7][8][34][35][36]. In the case of partitions of m into integers, it was only in relatively recent times that decades-old mathematical result have been exploited to calculate thermodynamic relations, namely for Bose-Einstein particles trapped in a one-dimensional harmonic potential [4,5,24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Gregorio

Rondoni

2018

Entropy

View full text Add to dashboard Cite

From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose-Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is 'large' and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of (2/3)-rds in one dimension, and (3/5)-ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.

show abstract

Section: Partitions and Compositions: The Case Of Integerssupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Gregorio

Rondoni

2018

Entropy

View full text Add to dashboard Cite

show abstract

“…An analytical solution for the BEC statistics at any temperature, including the entire critical region, was found recently [ 15 , 26 , 27 , 28 , 29 , 30 ] only for the ideal, non-interacting gas confined in a trap of arbitrary geometry and dimensions. In particular, it provides a universal scaling of all thermodynamic quantities and a universal form of the BEC occupation probability distribution over the entire critical region in the thermodynamic limit of macroscopically large systems.…”

Section: Critical Phenomena and Statistics Of Anomalous Fluctuatiomentioning

confidence: 99%

“…An essential novelty of the work [ 15 ] and the subsequent papers [ 26 , 27 , 28 , 29 , 30 ] was a formulation (and a solution) of the very problem of finding the entire probability distribution and its characteristic function for the BEC statistics as opposed to the calculation of just the first two statistical moments, the mean value and the variance, discussed in the preceding works (see, e.g., [ 11 , 13 , 14 , 18 , 23 ]). It was done mainly for the ideal Bose gas, with only one exception.…”

Section: Critical Phenomena and Statistics Of Anomalous Fluctuatiomentioning

confidence: 99%

Anomalous Statistics of Bose-Einstein Condensate in an Interacting Gas: An Effect of the Trap’s Form and Boundary Conditions in the Thermodynamic Limit

Tarasov

Kocharovsky

2018

Entropy

View full text Add to dashboard Cite

Abstract:We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions. Keywords: Bose-Einstein condensation; statistics of Bose-Einstein condensate; Bogoliubov coupling; interacting Bose gas; mesoscopic system Critical Phenomena and Statistics of Anomalous Fluctuations in a Bose-Einstein Phase TransitionAny second order phase transition in a many-body system occurs near its critical point via development of the critical phenomena and anomalously large fluctuations of an order parameter [1]. Finding their microscopic theory and analytical description constitutes one of the major and still unsolved problem in theoretical physics. A phase transition to the Bose-Einstein condensed state in the interacting Bose gas of N particles confined in an optical/magnetic trap is one of the phenomena that have been the most intensively studied in the last two decades (for a review, see [2] and references therein).In the present paper we analyze the statistics of the number of condensed particles at the temperature T well outside a narrow critical region around the critical temperature T c , namely, at T T c where a well-known mean-field approximation based on the Gross-Pitaevskii and Bogoliubov-de Gennes equations is valid. (In fact, such a mean-field approximation works reasonably well even up to the temperature T c /2, that is for all temperatures in the interval T < T c /2, as has been verified, for example, by the experiments on the dynamics and decay of the condensate excitations [2][3][4][5][6].) At somewhat higher temperatures, but still outside the critical region, one could employ a more complicated theory (beyond the mean field) that consistently accounts for a deeper thermal depletion of spatially inhomogeneous condensate and various many-body effects of particle interactions via an approximate solution to the Dyson equation for the Green's and self-energy functions of the standard quantum-field-theory method [3,4]. Even higher temperatures at the wings

show abstract

“…While for an ideal Bose gas there are differences between canonical (or micro-canonical) and grand-canonical predictions for many thermodnamic quantities (cf. [40,41]), they are particularly large for fluctuations of the number of particles in the ground state (cf. [26,29,30]).…”

Section: Introductionmentioning

confidence: 99%

Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light

Weiß

Tempere

2016

Phys. Rev. E

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.

show abstract

Grand Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of Bose–Einstein Condensation of an Ideal Gas in Arbitrary Trap

Cited by 14 publications

References 81 publications

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

Anomalous Statistics of Bose-Einstein Condensate in an Interacting Gas: An Effect of the Trap’s Form and Boundary Conditions in the Thermodynamic Limit

Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light

Contact Info

Product

Resources

About